A new paradigm for human stick balancing: a suspended not an inverted pendulum

Abstract

We studied 14 skilled subjects balancing a stick (a television antenna, 52 cm, 34 g) on their middle fingertip. Comprehensive three-dimensional analyses revealed that the movement of the finger was 1.75 times that of the stick tip, such that the balanced stick behaved more like a normal noninverted pendulum than the inverted pendulum common to engineering models for stick balancing using motors. The average relation between the torque applied to the stick and its angle of deviation from the vertical was highly linear, consistent with simple harmonic motion. We observed clearly greater rotational movement of the stick in the anteroposterior plane than the mediolateral plane. Despite this magnitude difference, the duration of stick oscillatory cycles was very similar in both planes, again consistent with simple harmonic motion. The control parameter in balancing was the ratio of active torque applied to the stick relative to gravitational torque. It determined both the pivot point and oscillatory cycle period of the pendulum. The pivot point was located at the radius of gyration (about the centre of mass) of the stick from its centre of mass, showing that the subjects attuned to the gravitational dynamics and mass distribution of the stick. Hence, the key to controlling instability here was mastery of the physics of the unstable object. The radius of gyration may—similar to centre of mass—contribute to the kinesthesis of rotating limb segments and control of their gravitational dynamics.

Appendices

Appendix 1: Moment of inertia (MoI) of the stick

The oscillation time of the stick was measured by suspending the stick at its base and averaging the period of pendulum sway over 10 cycles. This was repeated for three trials, and the average of the trials was used. This period was applied to Eq. 1 in order to calculate MoI (Tipler 1999):

$$ {\text{MoI}} = \frac{{T^{2} mgL}}{{4\pi^{2} }}\quad {\text{solving}}\,{\text{ for}}\,{\text{MoI}}\,{\text{using}}\,{\text{the}}\,{\text{period}} $$

(15)

where T is the period, m is the mass, g is the gravitational acceleration (9.8 m/s²), and L is the distance from the pivot point to the CoM. The MoI value derived empirically was checked theoretically. A stick was disassembled into the five sections of the telescopic antenna. By weighing the parts of the stick separately and applying the equations for composite moment of inertia of hollow cylinders, the theoretical MoI was calculated as shown below (Halliday et al. 2001):

$$ {\text{Theoretical}}\,{\text{MoI}} = \sum\limits_{n = 1}^{5} {{\text{MoI}}_{n} } \quad {\text{composite}}\,{\text{MoI}} $$

(16)

$$ {\text{MoI}}_{n} = {\text{MoI}}_{0} + m_{n} d_{n}^{2} \quad {\text{parallel}}\,{\text{axis}}\,{\text{theorem}} $$

(17)

$$ {\text{MoI}}_{0} = \frac{1}{12}m_{n} L_{n}^{2} \quad {\text{MoI}}\,{\text{of}}\,{\text{a}}\;{\text{rod}} $$

(18)

where n designates a dissembled section of the stick, m _n is the mass of the section, d _n is the distance from the pivot point to the CoM of the section, MoI _n is the MoI of the section pivoting at the base of the stick, MoI₀ is the MoI about the CoM of the section, and L _n is the length of the section. The two estimates of MoI derived empirically and theoretically were in close agreement (difference of empirical from theoretical value = 3.3 %), and so the MoI from the empirical value was used for subsequent analysis.

Appendix 2: Pendulum length, period and radius of gyration

We shall examine the relation between the length of a pendulum and its period of oscillation as described in Stephenson (1969), using the stick dynamics obtained from our results. Let the stick have a mass m and be suspended at O at a distance l from the CoM with a moment of inertia MoI _O about O. The pendulum length is expressed as L + l where L indicates the distance from the base to the CoM (Fig. 10). The general expression for the period of angular simple harmonic motion for a physical pendulum is then given by

$$ {\text{Period}}_{O} = 2\pi \sqrt {\frac{{{\text{MoI}}_{O} }}{mgl}} \quad ({\text{Stephenson}}\,1969). $$

(19)

Fig. 10

A physical pendulum showing the point of suspension (O), the distance (l) from the suspension point to the centre of mass (CoM) and the distance (L) from the base to the CoM. m mass of the stick, g gravitational acceleration

By the parallel axis theorem, the MoI _O may be written in terms of the MoI about the CoM (MoI_com) as

$$ {\text{MoI}}_{O} = {\text{MoI}}_{\text{com}} + ml^{2} . $$

(20)

The radius of gyration of the stick about the CoM, r _g , can be computed in terms of the moment of inertia about the CoM (MoI_com) and the mass m:

$$ {\text{MoI}}_{\text{com}} = mr_{g}^{2} . $$

(21)

Then,

$$ \begin{aligned} {\text{MoI}}_{O} & = {\text{MoI}}_{\text{com}} + ml^{2} \\ & = mr_{g}^{2} + ml^{2} \\ & = m(r_{g}^{2} + l^{2} ). \\ \end{aligned} $$

(22)

Therefore, Eq. 19 becomes

$$ {\text{Period}}_{O} = 2\pi \sqrt {\frac{{(r_{g}^{2} + l^{2} )}}{gl}} . $$

(23)

The relation between the period of oscillation and l in Eq. 23 is shown in Fig. 11. The period of the physical pendulum varies with the distance from the point of suspension to the CoM. The distance at which the period is a minimum is where the slope of the plot becomes 0. When differentiation is performed on Eq. 23, it is found that the slope becomes 0 when l = r _g . The minimum period of oscillation is thus

$$ {\text{Period}}_{\min } = 2\pi \sqrt {\frac{{2r_{g} }}{g}} . $$

(24)

Fig. 11

The period of oscillation of a physical pendulum plotted against the distance from the point of suspension to its centre of mass (CoM). The maximum period occurs when the pendulum is suspended at the CoM. The minimum period of oscillation (1.06 s) occurs when the pendulum is suspended at a distance of the radius of gyration (r _g ) from the CoM. Vertical arrows indicate, from left to right, the stick base, the CoM (0.27 m from base), the r _g from the CoM and the distance from the CoM to the stick tip

For the stick used in our experiment, we can calculate the moment of inertia MoI_tip ² about the tip of the stick, which is located at a distance l _tip from the CoM. Using the parallel axis theorem and Eq. 21, we get

$$ {\text{MoI}}_{\text{com}} = {\text{MoI}}_{\text{tip}} - ml_{\text{tip}}^{2} = mr_{g}^{2} $$

$$ \begin{aligned} r_{g} & = \sqrt {\frac{{{\text{MoI}}_{\text{tip}} - ml_{\text{tip}}^{2} }}{m}} \\ & = \sqrt {\frac{{{\text{MoI}}_{\text{tip}} }}{m} - l_{\text{tip}}^{2} } . \\ \end{aligned} $$

(25)

The MoI_tip value was calculated as 0.0028 kgm² for the stick with the axis of rotation at the tip. Since l _tip = 0.25 m is the distance between the CoM and the stick tip, the r _g value is calculated as

$$ r_{g} = \sqrt {\frac{0.0028}{0.0338} - 0.25^{2} } = 0.14\,{\text{m}}. $$

The pendulum length at which the period is a minimum then is L + r _g  = 0.27 m + 0.14 m = 0.41 m (see Figs. 1a, 11). This value, derived from the general expression for the period of angular simple harmonic motion of a physical pendulum, coincides with the pendulum lengths derived empirically from the magnitude values k of the ratio between the torque due to finger acceleration and the torque due to gravity (see Table 3).

In the scenario depicted in Fig. 11, only gravitational torque is exerted on the stick. In the case of the stick balancing, however, torque is actively exerted on the stick by the subject. Indeed, the magnitude ratio k between this active torque and the torque due to gravity is the major control variable available to the subject, and it determines the length of the pendulum. Hence, the pendulum length set by the torque applied by the subjects coincided with the pendulum length at which the period is minimum under gravitational torque alone.

Appendix 3: Period of oscillation of a pendulum under greater-than-gravity torque

Again using the stick dynamics obtained from our results, the minimum period of a physical pendulum under gravitational torque alone is derived by applying r _g to Eq. 24:

$$ {\text{Period}}_{\min } = 2\pi \sqrt {\frac{2 \cdot 0.14}{g}} = 1.06\,{\text{s}}. $$

The period of simple harmonic motion of a pendulum under greater-than-gravity torque can be calculated as follows. The expression for angular simple harmonic motion of a physical pendulum under gravity torque, specified by the constants m, g, L and MoI, is

$$ \ddot{\theta } = - \frac{mgL\theta }{\text{MoI}}.\quad ({\text{Stephenson}}\,1969). $$

(26)

The expression for angular simple harmonic motion of a physical pendulum under greater-than-gravity torque can be derived by combining Eqs. 10 and 13 and reorganising to get

$$ \ddot{\theta } = - \frac{(k - 1)mgL\theta }{\text{MoI}}. $$

(27)

Then, the period of simple harmonic motion under (k − 1) g torque can be calculated from Eq. 19, which becomes

$$ {\text{Period}}_{O} = 2\pi \sqrt {\frac{\text{MoI}}{mgL(k - 1)}} . $$

Using the mean value of k = 4.46 from our results, the period is

$$ 2\pi \sqrt {\frac{\text{MoI}}{mgL(k - 1)}} = 0.61\,{\text{s}}. $$

(28)

Greater-than-gravity torque reduces the period of a pendulum relative to that under gravitational torque alone. Thus, if this stick was suspended as a physical pendulum at a distance r _g from its CoM, its period of oscillation under the greater-than-gravity torque exerted by the subjects (0.61 s) would be shorter than its minimum period under gravitational torque alone (1.06 s). Any increase or decrease in the value of k, and hence in active torque applied by the subject, would, respectively, decrease or increase the period.

From the theoretical period value obtained under the greater-than-gravity torque measured from the subjects (Eq. 28), it can be seen that the half-cycle duration becomes 0.305 s, which fits closely with the values (0.30–0.33 s) measured independently from the kinematic data, shown in Table 2. Hence, the observed cycle duration is consistent with angular simple harmonic motion of the stick as a physical pendulum operating under greater-than-gravity torque.